This linear sequential estimator of mean and variance in Gaussian observations is called a Kalman Filter.

Note that the uncertainty about $\theta$ decreases over time (since $0<(1-K_t)<1$). This makes sense: since we assume that the statistics of the system do not change (stationarity), each new sample provides new information.

Let's implement the Kalman filter described above. We'll use it to recursively estimate the value of $\theta$ based on noisy observations. Use the 'Step' button to see the recursive updates to the posterior $p(\theta|D)$.